Theorem cotr3 15014

Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)

Hypotheses

Ref	Expression
cotr3.a	⊢ 𝐴 = dom 𝑅
cotr3.b	⊢ 𝐵 = (𝐴 ∩ 𝐶)
cotr3.c	⊢ 𝐶 = ran 𝑅

Assertion

Ref	Expression
cotr3	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr3

Step	Hyp	Ref	Expression
1		cotr3.a	. . 3 ⊢ 𝐴 = dom 𝑅
2	1	eqimss2i 4006	. 2 ⊢ dom 𝑅 ⊆ 𝐴
3		cotr3.b	. . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶)
4		cotr3.c	. . . . 5 ⊢ 𝐶 = ran 𝑅
5	1, 4	ineq12i 4179	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅)
6	3, 5	eqtri 2792	. . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅)
7	6	eqimss2i 4006	. 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
8	4	eqimss2i 4006	. 2 ⊢ ran 𝑅 ⊆ 𝐶
9	2, 7, 8	cotr2 15013	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∀wral 3085   ∩ cin 3912   ⊆ wss 3913   class class class wbr 5113  dom cdm 5662  ran crn 5663   ∘ ccom 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673

This theorem is referenced by: (None)